What Is A Conjecture

 

What is a Conjecture ?

 

 

Inductive  reasoning is a type of reasoning that allows you to reach conclusions  based on a pattern of a specific examples or past events. A conclusion  reached by using inductive reasoning is called  CONJECTURE.

 

For each  sequence below, find the next term by first finding a pattern. Then using a pattern, make a conjecture about the next term.

 

 

Example number one :

 

Given sequence  :  1, 8, 27, 64 ,  ____

 

Conjecture  :   Y   =   X^3

 

Verifying Conjecture :

 

Let X  = 1     then   Y  =  1^3    =   1

Let X =  2     then   Y  =   2^3    =   8

Let  X  = 3    then   Y  =   3^3    =   27

Let   X  = 4   then   Y  =   4^3    =   64

 

What is the next number in the series ?

 

Let  X  =  5  then   Y  =   5^3  =    125

The next number in the series is 125.

 

 

Example  number  two  :

 

Given sequence  :   3, 5,  11,  29,  ____

 

Conjecture  :  Y  =   3^X  +   2

 

Verifying Conjecture :

 

Let  X  =  0  then   Y  = 3^0 +  2   =  1  + 2  =  3

Let X  =  1    then   Y  =  3^1 + 2  =  3   +  2  = 5

Let  X  = 2  then    Y =  3^2 +  2   =  9   +  2  =  11

Let  X  =  3  then   Y  =  3^3  +  2  =  27  + 2  = 29

 

What is the next number in the series  ?

 

Let  X  =  4  then   Y  =   3^4   +  2    =  81  +  2   =  83

The next number in the series is  83.

 

 

Example number three :

 

Given  sequence :  3, 5, 7, 9,  ___

 

Conjecture   :   Y  =  2X  +  3

 

Verifying Conjecture :

 

Let  X  = 0  then  Y  =  2(0)  +  3   =   3

Let  X  =  1  then  Y  = 2(1)  +  3  =    5

Let  X  =  2  then  Y  = 2(2)   +  3   =  7

Let  X  =  3  then   Y  =  2(3)  + 3   =  9

 

What is the next number in the series ?

 

Let  X  =  4  then    Y   =  2(4)  +   3  =  11

 

The next number in the series is  11

 

For your practice exercises try to find the conjecture of the following series:

 

(1)    5, 8, 11,  14

(2)    1,  4, 16,  64

(3)    2, 3.  5,   9

(4)    3, 6, 9, 12

(5)    1, 2. 4. 8. 16

 

 

 Answers  :

 

1)      Y =  3X  +  5

2)      Y = 2 ^2X

3)      Y = 2^X  +  1

4)      Y  =  3X

5)      Y =  2^X

 

 

Reference : GEOMETRY  by  Neo Asia Publishing

 

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